8.3. Steady Aerodynamics#
todo
Some remarks, using vorticity dynamics:
the starting vortex: when were vorticity and circulation generated?
vortex intensity of the wake: \(\boldsymbol\gamma_w(\mathbf{r}_w(\mathbf{r}_TE)) = \Delta \boldsymbol\gamma(\mathbf{r}_TE)\), from integral balance of vorticity in a stream-tube (todo check it and add details)
wake dynamics: transport of vorticity + vortex stretching; using vorticity equation, the shape of the wake \(\mathbf{r}_w(\mathbf{r}_TE)\) and its intensity \(\gamma(\mathbf{r}_w)\) can be computed (todo check it and add details) - or use Helmholtz’s vortex theorem + Kelvin’s circulation theorem (todo Add sections in the vorticity dynamics, or add a section here in Aerodynamics chapter, under the assumptions of almost-everywhere irrotational flow and negliglible viscosity effects.)
8.3.1. Mathematical model#
In order to build a domain where the flow is irrotational, wake regions must be cut from the domain: both sides of these regions become part of the boundary of the domain. As shown below, there’s no physical wake for 2-dimensional steady flows, while wakes exist in 3-dimensional flows.
8.3.2. Green’s function method#
Derivation of the Green’s function for the Poisson problem
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Solution.
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\(E(\mathbf{r}_0)\)
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Asymptotic behavior of the perturbation velocity
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Doublet/vortex equivalence
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Zero wake vorticity in steady 2-dimensional problems
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Details in integration by parts on surface in doublet/vortex equivalence
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Relationship between doublet/vortex distribution and the circulation - 2-dimensional steady problems
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\(\oint_{\mathbf{r}_0 \in \ell_0} \hat{\mathbf{t}}_0 \times \nabla_0 G(\mathbf{r}; \mathbf{r}_0)\)
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Asymptotic behavior of the perturbation potential
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8.3.3. Wake and the shape of the domain#
Physical conditions, with jump conditions
No physical wake in 2-dimensional steady flows
Wake in steady 2-dimensional flows
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